cholesky_banded(ab, overwrite_ab=False, lower=False, check_finite=True)
The matrix a is stored in ab either in lower-diagonal or upper- diagonal ordered form
ab[u + i - j, j] == a[i,j] (if upper form; i <= j)
ab[ i - j, j] == a[i,j] (if lower form; i >= j)
Example of ab (shape of a is (6,6), u=2)
upper form:
* * a02 a13 a24 a35
* a01 a12 a23 a34 a45
a00 a11 a22 a33 a44 a55
lower form:
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 * *
Banded matrix
Discard data in ab (may enhance performance)
Is the matrix in the lower form. (Default is upper form)
Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.
Cholesky factorization of a, in the same banded format as ab
Cholesky decompose a banded Hermitian positive-definite matrix
cho_solve_banded
import numpy as np
from scipy.linalg import cholesky_banded
from numpy import allclose, zeros, diag
Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]])
A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1)
A = A + A.conj().T + np.diag(Ab[2, :])
c = cholesky_banded(Ab)
C = np.diag(c[0, 2:], k=2) + np.diag(c[1, 1:], k=1) + np.diag(c[2, :])
np.allclose(C.conj().T @ C - A, np.zeros((5, 5)))
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.linalg._decomp_cholesky:cho_solve_banded
scipy.interpolate._bspl:_norm_eq_lsq
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